. Volume of a solid obtained by revolving area about an axis Definite integrals have applications in finding volumes of solids of revolution about a fixed axis. By a solid of revolution about a fixed axis, we mean that a solid is generated when a plane region in a given plane undergoes one full revolution about a fixed axis in the plane. For instance, consider the semi circular plane region inside the circle x and above the x -axis.
See Fig. . . If this region is given one complete rotation (revolution for ° = π radians) about x -axis, then a solid called a sphere is generated.
In the same manner, if you want to generate a right-circular cylinder with radius a and height h , you can consider the rectangular plane region bounded between the straight lines y = , y , x = and x h in the xy -plane. See Fig. . .
If this region is given one complete rotation (revolution for ° = π radians) about x -axis, then a solid called a cylinder is generated. We restrict ourselves to obtain volume of solid of revolution about x -axis or y -axis. Whenever solid of revolution about x -axis is considered, the plane region that is revolved about x -axis lies above the x -axis. So, in this region y ≥ .
Whenever solid of revolution about y -axis is considered, the plane region that is revolved about y -axis lies to the right of y -axis. So, in this region x ≥ . We shall find the formula for finding the volume of the solid of revolution of the plane region in the first quadrant bounded by the curve y ( ) , x -axis and the lines x and x > about x -axis. The derivation of the formula is based upon the formula that the volume of a cylinder of radius r and the height h is p r h .
Assume that every line parallel to y- axis lying between the lines x and x > cuts the curve y ( ) in the first quadrant exactly at one point. Divide a b